16 Invariant speed and local conservation

In Newton’s theory, gravitational effects are simultaneous with their causes: the Sun attracts the Earth towards the Sun’s present position. This is often seen as the reason why Newton was in no position to “frame hypotheses” (about the mechanism or natural process by which gravity acts).

Electromagnetic effects, on the other hand, are retarded. The earliest time at which a solar flare can affect us is about eight minutes later — the distance between the Sun and the Earth divided by the so-called speed of light (c). According to a widely held belief, the retardation of electromagnetic effects made it possible to explain how — by what mechanism or natural process — electric charges act on electric charges.

Although we have previously (here and here) disposed of such an “explanation” as a mere sleight-of-hand, it is worth taking a look at what changed and what did not change when Einstein realized that the invariant speed was finite (namely, c) rather than infinite, as Newton had held. (Reminder: anything that “travels” with this speed in one inertial frame, does so in every other inertial frame.)

The existence of an invariant speed implies a special kind of spatiotemporal relation between events: either the relation of being simultaneous, which is absolute (that is, independent of the inertial frame used) in Newton’s (non-relativistic) theory, or the relation of being situated on each other’s light cone, which is absolute in Einstein’s (relativistic) theory.

Suppose that an event e1 at (x1,t1) is the cause of an event e2 at (x2,t2). The fact that e2 happens at t2, rather than at any other time, has two possible explanations. If the action of e1 on e2 is mediated, t2 is determined by the speed of mediation. This could be the speed of a material object traveling from (x1,t1) to (x2,t2), the speed of a signal propagating in an elastic medium, or what have you. But if the action of e1 on e2 is unmediated, t2 is determined by the special kind of spatiotemporal relation that the existence of an invariant speed implies. In Newton’s theory, in which simultaneity is absolute, t2 is equal to t1, whereas in the relativistic theory, t2 is retarded by |x2–x1|/c. So if an effect e2 at x2 happens a time span Δt = |x2–x1|/c after its cause e1 at x1, it means that the effect that e1 has on e2 is unmediated; by no means does it follow that e2 is brought about through the mediation of something that travels from e1 to e2 with the invariant speed c.

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Noether’s theorem

Spacetime coordinates, we said, are human inventions. To make the laws of physics as simple as possible, we introduce them in such a way that equal coordinate intervals are physically equivalent. This means, among other things, that freely moving classical particles travel equal space intervals Δx,Δy,Δz in equal time intervals Δt; the ratios formed of Δx, Δy, Δz, and Δt are constants. The physical equivalence of equal time intervals implies a conserved physical quantity — energy; and the physical equivalence of equal intervals of the space coordinates implies another conserved physical quantity — momentum.

These results have been generalized by Noether’s theorem. Suppose that we have a theory that is defined by a Lagrangian L, and suppose that L is invariant under some continuous transformation of the fields on which it depends. Noether’s theorem then implies a locally conserved quantity Q. This means that for any region R of space, the total amount of Q inside R increases (or decreases) by the amount of Q that flows into R (or out of R) through the boundary of R.

If, for instance, L is invariant under translations in space (which it can only be if equal space intervals are physically equivalent), then the theorem implies the local conservation of momentum, and if L is invariant under time translations (which it can only be if equal time intervals are physically equivalent), then the theorem implies the local conservation of energy. More compactly, if L is invariant under spacetime translations, Noether’s theorem implies the local conservation of energy-momentum.

A gauge transformation is another continuous transformation of the fields on which L depends. If L is invariant under such a transformation, the locally conserved quantity implied by this invariance is charge — electric charge in the case of this gauge transformation, or the weak (or flavor) charge associated with particles interacting via the weak force, or the strong (or color) charge associated with particles interacting via the strong force.

So are we now to imagine that energy, momentum, and charge are kinds of stuff that continuously “slosh around” in space or spacetime? Of course not. The local conservation laws, like the Lagrangians that imply them, are calculational tools. They ensure that, for every scattering event and for every inertial frame, the energies, momenta, and charges of the incoming particles equal the energies, momenta, and charges (respectively) of the outgoing particles. (If some energy–momentum escapes undetected, then it also warrants the following conditional: if the escaped energy–momentum were detected, it would agree with the local conservation law for energy-momentum.)